Jose Arnaldo B. Dris

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 23, 2017, Number 4, Pages 1—13

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## Details

### Authors and affiliations

Jose Arnaldo B. Dris

*University of the Philippines-Diliman
*

### Abstract

We investigate the implications of a curious biconditional involving the divisors of odd perfect numbers, if Dris conjecture that *q ^{k}* <

*n*holds, where

*q*

^{k}n^{2}is an odd perfect number with Euler prime

*q*. We then show that this biconditional holds unconditionally. Lastly, we prove that the inequality

*q*<

*n*holds unconditionally.

### Keywords

- Odd perfect number
- Abundancy index
- Deficiency

### AMS Classification

- 11A25

### References

- Acquaah, P. & Konyagin, S. (2012) On prime factors of odd perfect numbers, Int. J. Number Theory, 8(6), 1537–1540.
- Beasley, B. D. (2013) Euler and the ongoing search for odd perfect numbers, ACMS 19th Biennial Conference Proceedings, Bethel University, May 29 to Jun. 1, 2013.
- Brown, P. A. (2016) A partial proof of a conjecture of Dris, preprint, http://arxiv. org/abs/1602.01591.
- Dagal, K. A. P. & Dris, J. A. B. (2017) The abundancy index of divisors of odd perfect numbers – Part II, preprint, https://arxiv.org/abs/1309.0906.
- Dris, J. A. B. (2008) Solving the Odd Perfect Number Problem: Some Old and New Approaches, M. Sc. Math thesis, De La Salle University, Manila, Philippines.
- Dris, J. A. B. (2012) The abundancy index of divisors of odd perfect numbers, J. Integer Seq., 15, Article 12.4.4.
- Dris, J. A. B. (2017) Conditions equivalent to the Descartes–Frenicle–Sorli conjecture on odd perfect numbers, Notes on Number Theory and Discrete Mathematics, 23(2), 12–20.
- Dris, J. A. B. (2017) New results for the Descartes–Frenicle–Sorli conjecture on odd perfect numbers, Journal for Algebra and Number Theory Academia, 6(3), 95–114.
- Woltman, G., & Kurowski, S. (2017) The Great Internet Mersenne Prime Search, http: //www.mersenne.org/primes/. Last viewed: August 12, 2017.
- Holdener, J. A. (2006) Conditions equivalent to the existence of odd perfect numbers, Math. Mag., 79 (5), 389–391.
- Ochem, P. & Rao, M. (2012) Odd perfect numbers are greater than 101500, Math. Comp., 81(279), 1869–1877.
- Sloane, N. J. A. OEIS sequence A033879 – Deficiency of n, or 2n–σ(n), http://oeis. org/A033879.
- Starni, P. (2017) On Dris conjecture about odd perfect numbers, preprint, https:// arxiv.org/abs/1706.02144.
- Sorli, R. M. (2003) Algorithms in the Study of Multiperfect and Odd Perfect Numbers, Ph. D. Thesis, University of Technology, Sydney, http://hdl.handle.net/10453/ 20034.

## Related papers

- Dris, J. A. B., & San Diego, I. T. (2020). Some modular considerations regarding odd perfect numbers – Part II. Notes on Number Theory and Discrete Mathematics, 26 (3), 8-24, doi: 10.7546/nntdm.2020.26.3.8-24.

## Cite this paper

APADris, J. A. B. (2017). On a Curious Biconditional Involving the Divisors of Odd Perfect Numbers. Notes on Number Theory and Discrete Mathematics, 23(4), 1-13.

ChicagoDris, Jose Arnaldo B. “On a Curious Biconditional Involving the Divisors of Odd Perfect Numbers” Notes on Number Theory and Discrete Mathematics 23, no. 4 (2017): 1-13.

MLADris, Jose Arnaldo B. “On a curious biconditional involving the divisors of odd perfect numbers.” Notes on Number Theory and Discrete Mathematics 23.4 (2017): 1-13. Print.